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Theorem uspgrushgr 29034
Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
Assertion
Ref Expression
uspgrushgr (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Proof of Theorem uspgrushgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2725 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isuspgr 29009 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4 ssrab2 4069 . . . . 5 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5 f1ss 6794 . . . . 5 (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
64, 5mpan2 689 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
73, 6biimtrdi 252 . . 3 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
81, 2isushgr 28918 . . 3 (𝐺 ∈ USPGraph → (𝐺 ∈ USHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
97, 8sylibrd 258 . 2 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph))
109pm2.43i 52 1 (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  {crab 3419  cdif 3936  wss 3939  c0 4318  𝒫 cpw 4598  {csn 4624   class class class wbr 5143  dom cdm 5672  1-1wf1 6540  cfv 6543  cle 11279  2c2 12297  chash 14321  Vtxcvtx 28853  iEdgciedg 28854  USHGraphcushgr 28914  USPGraphcuspgr 29005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5301
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551  df-ushgr 28916  df-uspgr 29007
This theorem is referenced by:  uspgrupgrushgr  29036  usgredgedg  29087  vtxdusgrfvedg  29349  1loopgrvd2  29361
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